Question: a . In a given undirected graph G , an independent set of G is any subset S of nodes in G such that there

a. In a given undirected graph G, an independent set of G is any subset S of nodes in G such that there are no edges between the nodes of S. The independent set problem is the problem of finding a maximum-size independent set in an input undirected graph G. Give a greedy algorithm for this problem, and prove by a counter example of a graph of at least 7 nodes that the Greedy solution is not always optimal.
b. In a given undirected graph G, a vertex cover of G is any subset C of nodes in G such that every edge in G involves at least one node in C. The vertex cover problem is the problem of finding a minimum-size vertex cover in an input undirected graph G. Give a greedy algorithm for this problem, and prove that the Greedy solution is not always optimal using a counter example of a graph of at least 7 nodes.

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